Every non-trivial finite solvable group has a normal subgroup of prime index

Question

Let $G$ be a non-trivial finite solvable group.Then $G$ has a normal subgroup of prime index.

I use the below fact to solve the problem.

"Every finite solvable group has a composition series that every factor of the series is cyclic of prime order."

I wonder if there is any other way to solve the problem.

Answer 1

Hint: choose a maximal normal subgroup $M$. Then $G/M$ is simple and is solvable, hence cyclic of prime order.